The Iterative Simplicity of Basic Topological Operations

TL;DR

This paper investigates the algebraic structures formed by basic topological operations like closure, interior, and boundary, focusing on their finiteness and noncommutative properties, and explores whether these semigroups are always finite.

Contribution

It introduces the study of semigroups generated by fundamental topological operations and examines their finiteness and algebraic properties, highlighting open problems.

Findings

Some semigroups are finite and noncommutative

Open question on whether all such semigroups are finite

Provides a framework for analyzing algebraic structures in topology

Abstract

Semigroups generated by topological operations such as closure, interior or boundary are considered. It is noted that some of these semigroups are in general finite and noncommutative. The problem is formulated whether they are always finite.